{"title":"Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials","authors":"Joseph P.S. Kung","doi":"10.1016/j.aam.2025.102866","DOIUrl":null,"url":null,"abstract":"<div><div>We present two sufficient conditions on a rank-<em>r</em> coloop-free matroid <em>M</em> for the additive Merino-Welsh inequality to hold. The first is that the density of <em>M</em> is sufficiently large. The second is that all the cocircuits of <em>M</em> have at least <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> elements. These results are “inconsequential” in the sense that although they show that a version of the conjecture holds for many matroids, they are far from covering all the possible cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102866"},"PeriodicalIF":1.3000,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0196885825000284","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
We present two sufficient conditions on a rank-r coloop-free matroid M for the additive Merino-Welsh inequality to hold. The first is that the density of M is sufficiently large. The second is that all the cocircuits of M have at least elements. These results are “inconsequential” in the sense that although they show that a version of the conjecture holds for many matroids, they are far from covering all the possible cases.
期刊介绍：
Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas.
Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.
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